From Euclidean to Minkowski space with the Cauchy-Riemann equations

TL;DR

This paper introduces a novel method using Cauchy-Riemann equations for analytically continuing Euclidean Green's functions to Minkowski space in non-perturbative quantum field theory, addressing uncertainties in dispersive approaches.

Contribution

It proposes an elementary, local method for analytical continuation that does not require global knowledge of the function, applied to the quark propagator in QCD.

Findings

The method can perform the continuation using lattice and Dyson-Schwinger data.

It faces challenges due to instability from high-frequency noise.

The approach offers new insights into Wick rotation and analytical structures.

Abstract

We present an elementary method to obtain Green's functions in non-perturbative quantum field theory in Minkowski space from calculated Green's functions in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes is many times unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore we suggest to use the Cauchy-Riemann equations, that perform the analytical continuation without assuming global information on the function in the entire complex plane, only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge Quantum Chromodynamics, that is known from lattice and Dyson-Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy-Riemann equations to high-frequency noise, that makes difficult to achieve good accuracy. We also point out a few curiosities related to the Wick rotation.